In fact, by [1]} we have
\(\mathrm {Cl}_2(L_{3, d})\)  is cyclic non-trivial if and only if \(d\)  has one of the following forms:

\(d= q\equiv 7\pmod {8}\)  is a prime integer.

\(d=qp\) , where \(q\equiv 3\pmod {8}\)  and \(p\equiv 5\pmod {8}\)  are prime integers.

Since \(rank_2(\mathrm {Cl}_2(L_{m, d}))\ge rank_2(\mathrm {Cl}_2(L_{3, d})), \)  then we get the result by the previous theorem.
